Sine, Cosine and Tangent

Naming the sides from an angle

Take a right triangle and choose one of its two acute angles. The angle is usually called $\theta$, the Greek letter theta. The three sides are then named by where they sit in relation to $\theta$.

The hypotenuse is the side across from the right angle. It is the longest side, as Pythagoras’ theorem showed. The opposite side is the side across from $\theta$: it is the one side that does not touch $\theta$. The adjacent side is the other side that touches $\theta$, the one that is not the hypotenuse. Adjacent means next to.

The names depend on the angle you choose. For the other acute angle, the opposite and adjacent sides swap over. The hypotenuse stays the same, because it is named from the right angle.

Seen from the angle $\theta$, marked by the arc: the opposite side is across from it, the adjacent side runs from it to the right angle, and the hypotenuse is across from the right angle.

A ratio that depends only on the angle

Here is a right triangle with sides 3, 4 and 5, and $\theta$ is the angle across from the side of 3. The opposite side is 3 and the hypotenuse is 5, so opposite ÷ hypotenuse $= \frac{3}{5} = 0.6$.

Now double every side, to 6, 8 and 10. The angles do not change, and opposite ÷ hypotenuse $= \frac{6}{10} = 0.6$ again. Triple them, to 9, 12 and 15, and it is $\frac{9}{15} = 0.6$.

This happens for every right triangle with the same angle $\theta$. Two such triangles both have a right angle and the angle $\theta$, so their third angles are equal too, and by the AA test they are similar. One is an enlargement of the other by some scale factor k, so its opposite side is k times as long and its hypotenuse is k times as long. The k cancels when one is divided by the other, and the ratio is the same.

So opposite ÷ hypotenuse depends only on the angle, not on the size of the triangle. Because each angle has its own fixed value, the ratio is given a name: the sine of $\theta$, written $\sin \theta$. In the 3-4-5 triangle, $\sin \theta = \frac{3}{5}$.

A right triangle with $\theta = 30^\circ$ and a hypotenuse of 2. Pull the corner outward: the triangle grows, and the three ratios on the right stay exactly the same, because every side is multiplied by the same number. Swing the corner round: the angle changes, and so do the ratios. Swing it to 45° and read them there.

Cosine and tangent

The same argument works for any two sides, so each pair gives a ratio of its own. Adjacent ÷ hypotenuse is the cosine of $\theta$, written $\cos \theta$. Opposite ÷ adjacent is the tangent of $\theta$, written $\tan \theta$. The tangent is the one ratio that does not use the hypotenuse.

In the 3-4-5 triangle, the adjacent side is 4, so $\cos \theta = \frac{4}{5} = 0.8$ and $\tan \theta = \frac{3}{4} = 0.75$.

The three together, in letters: $\sin \theta =$ opposite/hypotenuse, $\cos \theta =$ adjacent/hypotenuse, and $\tan \theta =$ opposite/adjacent. Many students remember them with the word SOHCAHTOA: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent.

The tangent is also the sine divided by the cosine, because the hypotenuse cancels: $\frac{3}{5} \div \frac{4}{5} = \frac{3}{4}$. That is a quick check on three ratios worked out from one triangle.

The 3-4-5 triangle, with $\theta$ across from the side of 3: $\sin \theta = \frac{3}{5}$, $\cos \theta = \frac{4}{5}$ and $\tan \theta = \frac{3}{4}$.

The other acute angle

In the same triangle, look from the other acute angle, the one at the top. The side of 4 is across from it, so 4 is now the opposite side and 3 is the adjacent side. For that angle, the sine is $\frac{4}{5}$, the cosine $\frac{3}{5}$ and the tangent $\frac{4}{3}$.

So always name the sides from the angle in the question before writing a ratio. Notice that the sine of one acute angle is the cosine of the other: both are $\frac{4}{5}$.

How big the ratios can be

The hypotenuse is the longest side, so the opposite and adjacent sides are both shorter than it. That makes $\sin \theta$ and $\cos \theta$ always less than 1 in a right triangle. A sine of $\frac{5}{3}$ would mean an opposite side longer than the hypotenuse, which cannot happen.

The tangent has no such limit. It is less than 1 when the opposite side is shorter than the adjacent side, exactly 1 when the two are equal, and more than 1 when the opposite side is longer.

On a circle of radius 1

Put the angle $\theta$ at the center of a circle of radius 1, with one arm along the horizontal. The other arm meets the circle at a point. Drop a line from that point straight down to the horizontal, and there is a right triangle whose hypotenuse is the radius, 1.

With a hypotenuse of 1, $\sin \theta =$ opposite ÷ 1 = the opposite side, which is the height of the point. And $\cos \theta =$ adjacent ÷ 1 = the adjacent side, which is how far across the point is from the center.

This is the same sine that gives the area of a segment. There, $\sin \theta$ was the height of the triangle when the radius is 1, and on a radius of r the height was $r \sin \theta$. That height and the radius make a right triangle with the angle $\theta$, and opposite ÷ hypotenuse $= r \sin \theta / r = \sin \theta$. The two descriptions give the same number.

A right triangle only holds acute angles, less than 90°. On the circle the arm can keep turning past 90°, and the height and the distance across still have values there. Looking ahead, that is how the sine and cosine of larger angles are defined.

A radius of 1 at 37°. The gold height is sin 37° = 0.60 and the distance across is cos 37° = 0.80, to two decimal places, close to the $\frac{3}{5}$ and $\frac{4}{5}$ of the 3-4-5 triangle, whose angle is about 36.9°. Drag the point round: the height grows to 1 at 90°, where sin 90° = 1.

On a calculator

A calculator gives the three ratios for any angle with its sin, cos and tan keys, set to degrees. For example, sin 37° = 0.6018, cos 37° = 0.7986 and tan 37° = 0.7536, each to four decimal places.

The usual mistakes

Mixing up sine and cosine. The sine uses the side across from the angle and the cosine uses the side next to it. In the 3-4-5 triangle, $\frac{4}{5}$ is $\cos \theta$, not $\sin \theta$.

Putting the hypotenuse into a tangent. The tangent compares the two shorter sides, opposite over adjacent: $\frac{3}{4}$, not $\frac{3}{5}$.

Naming the sides from the wrong angle. Opposite and adjacent swap for the other acute angle, so find the angle in the question first.

Turning a ratio upside down. A sine or cosine with the hypotenuse on top, such as $\frac{5}{3}$, is more than 1, and no sine or cosine of an angle in a right triangle can be.

A ratio that gives a side

Because a ratio is fixed by the angle, it can give a side you have not measured. Choose the ratio that uses the side you know and the side you want. If the angle has $\sin \theta = \frac{3}{5}$ and the hypotenuse is 10, the opposite side x makes $\frac{x}{10} = \frac{3}{5}$, so x = 6.

In the application below, a kite string is let out at the same angle. The sine pairs the height with the string, and the tangent pairs the height with the distance along the ground.